Sunday, 23 June 2013

Strange Beliefs about Abstract Objects

Occasionally, the strange belief that mathematical objects are physical objects is advocated. I am truly baffled when I hear such beliefs. Here are the some questions:

Is the number $0$ a physical object?Is the number $2^{2^{2^{2^{2^{2^{2^{2}}}}}}}$ a physical object?Is the wellorder $(\omega, <)$ a physical object?Is the topological space $\mathbb{R}^4$ a physical object?Is the Lie group $SU(3)$ a physical object?Is the rank $V_{\omega + 57}$ a physical object?

For example, what is the mass of $(\omega, <)$? Can you find it somewhere, perhaps at Tesco's?

They insist, e.g., that because a physical computer, say, is finite, there aren't numbers beyond what it represents. But this philosophical conclusion requires the further assumption that numbers *are* physical objects. This assumption is the one that is not justified.

Another example - the physicality of mathematical objects is advocated by Doron Zeilberger:

"(ii) the traditional real line is a meaningless concept. Instead the real REAL ‘line’, is neither real, nor a line. It is a discrete necklace! In other words R = hZp, where p is a huge and unknowable (but fixed!) prime number, and h is a tiny, but not infinitesimal , ‘mesh size’. Hence even the potential infinity is a meaningless concept."